3. Draw a tangent to a circle, parallel to a given straight line.

4. Draw, in a circle, a chord of given length, passing through a given point.

5. Draw, in a circle, a chord of given length, and parallel to a given straight line.

6. Between two parallels draw a straight line of given length passing through a given point.

7. Between two intersecting straight lines draw a straight line of given length, and parallel to a given straight line.

8. With a given radius describe a circle which shall pass through two given points.

9. With a given radius describe a circle which shall pass through a given point, and be tangent to a given straight line.

10. With a given radius describe a circle which shall pass through a given point, and be tangent to a given circle.

II. With a given radius describe a circle tangent to two given straight lines.

12. With a given radius describe a circle tangent to a given straight line and given circle.

13. With a given radius describe a circle tangent to two given circles.

14. Describe a circle which shall cut three equal chords of given length from three given straight lines.

15. With a given radius describe a circle which shall be at the same given distance from three given points not in the same straight line.

16. Inscribe a circle in a given rhombus.

17. Find a point on a given straight line, at equal distances from two given points.

18. Trisect a right angle.

19. Trisect a given straight line.

20. From the vertices of a triangle as centres, describe three circles which shall touch each other, two and two.

21.

Find how many circles may be constructed equal to a given circle, touching it, and tangent to each other, two and two.

CONSTRUCTION OF TRIANGLES, ETC.

22. Construct an isosceles triangle, given its base and adjacent angle.

23. Construct an isosceles triangle, given its base and the radius of the inscribed circle.

24. Construct a right angled triangle, given the hypothenuse and one of the acute angles.

25. Construct a right angled triangle, given the hypothenuse and the perpendicular let fall from the right angle on the hypothenuse.

26. Construct a right angled triangle, given the hypothenuse and the radius of the inscribed circle.

27. Construct a triangle, given one side, an adjacent angle, and the sum of the two other sides.

28. Construct a triangle, given one side, an adjacent angle, and the difference of the two other sides.

29. Construct a right angled triangle, given the radius of the inscribed circle, and the radius of the circumscribed circle.

30. Construct a right angled triangle, given the radius of the inscribed circle, and one of the acute angles.

31. Construct a right angled triangle, given the radius of the inscribed circle, and one of the sides containing the right angle.

32. Construct a right angled triangle, given the median and altitude drawn from the vertex of the right angle.

33. Construct a triangle, given two sides and one altitude (two problems).

34. Construct a triangle, given two sides and one median line (two problems).

35. Construct a triangle, given the angles and the radius of the circumscribed circle.

36. Construct a triangle, given the three medians.

37. Construct a triangle, given the middle points of the three sides.

38. Construct a triangle, given two vertices; and, first, the point of intersection of the medians; second, the point of intersection of the three altitudes; third, the point of intersection of the three bisectrices of the angles.

39. Construct a triangle, given one vertex and the feet of two , altitudes.

40. Construct a triangle, given the feet of the three altitudes.

41. Construct a triangle, given the centres of the three escribed circles.

42. Construct a triangle, given the radius of the inscribed circle, the radius of an escribed circle, and one angle.

43. Construct a triangle, given the angles and one altitude.

44. Construct a square, given its diagonal.

45. Construct a rhombus, given its two diagonals.

46. Construct a rectangle, given the diagonal and one side.

47. Construct a square, given the sum of its diagonal and side.

48. Construct a square, given the difference of its diagonal and side.

49. Construct a trapezoid, given its four sides, and it being stated, also, which two are parallel.

50. Construct a pentagon, given the middle points of its sides.

51. Construct a rectangle, given its perimeter and its diagonal.

52. Construct a rhombus, given one side and the sum of its diagonals.

53. Construct a rhombus, given one of its angles and the radius of the inscribed circle.

BOOK III.

THE PROPORTIONS OF FIGURES.

DEFINITIONS.

1. Those figures whose surfaces are equal, we shall call equivalent figures.

Two figures may be equivalent, although very dissimilar: for example, a circle may be equivalent to a square, a triangle to a rectangle, etc.

:

The denomination of equal figures we shall reserve for such as when applied the one to the other, coincide in all their points of this kind are two circles the radii of which are equal, two triangles of which the three sides are equal, each to each, etc.

2. Two figures are similar when they have their corresponding angles equal, each to each, and their homologous sides proportional. By homologous sides are meant those which have a corresponding position in the two figures, or which are adjacent to equal angles. These angles themselves are called homologous angles.

Two equal figures are always similar; but two similar figures may be very unequal.

3. In different circles similar arcs, similar sectors, similar segments are those which correspond to equal angles at the centre.

Thus, if the angle A is equal to the angle O, the arc BC will be similar to the arc DE, the sector ABC to the sector ODE, etc.

ла

B

4. The altitude of a parallelogram is the perpendicular, EF, which measures the distance between the two opposite sides, AB, CD, taken as bases.

D E

B

5. The altitude of a triangle is the perpendicular, AD, let fall from the vertex of an angle, A, on the opposite side, BC, taken as a base.

B

D

D E C

6. The altitude of a trapezoid is the perpendicular, EF, drawn between its two parallel sides, AB, CD.

B

7. The area and the surface of a figure are terms which are nearly synonymous. Area designates more particularly the superficial magnitude of a figure, in so far as it is measured, or compared, with other surfaces.

N. B. In order to understand this and the following books, it will be necessary to bear in mind the theory of proportions, for which we refer to the introduction. [See Introduction.] We shall make only one observation, which is very important for fixing the true import of propositions, and for dissipating all obscurity, whether in their enunciation or their demonstration.

If we have the proportion A: B :: C: D, it is known that the product of the extremes, Ax D, is equal to the product of the means, Bx C.

This truth is incontestible for numbers: we add that it is equally so for magnitudes of any kind whatever, provided they be expressed, or we may imagine them expressed, in numbers; and this may always be supposed. For example, if A, B, C, D are lines, we may conceive that one of these four lines, or a fifth, if requisite, serves for a common measure for them all, and that it is taken for unity; then A, B, C, D will each represent a certain number of units, entire or fractional, commensurable or incommensurable, and the proportion between the lines A, B, C, D becomes a numerical proportion.

The product of the lines A and D, which is also named their rectangle, is then nothing else than the number of linear units contained in A, multiplied by the number of linear units contained in B; and it is easily conceived that this product can be, and must be, equal to that which results similarly from the lines B and C.

The magnitudes A and B may be of one kind, for example, lines; and the magnitudes C and D of another kind, for example, surfaces; in this case it will be necessary always to regard these magnitudes as numbers: A and B will be expressed in linear units, C and D in